Abstract

We report the experimental implementation of a new method for generating multiple dynamical optical tweezers, where each one of them is generated with an independent linear polarization state with arbitrary orientation. This also allows an independent simultaneous polarization-rotation control. The laser beam, both for generating multiple traps and polarization control, has been modulated using a single reflective nematic liquid crystal with parallel alignment. We present experimental results of controlled displacement, orientation and rotation of birefringent particles. In addition, a simple method for estimating and canceling out the primary astigmatism present in the system is presented.

]. This reference shows the rotation of CuO particle trapped due to the helical phase structure of the beam. The Laguerre-Gaussian (LG) mode has been used to transfer orbital angular momentum (OAM) to a trapped particle in an optical tweezer setup [3

], and the cancellation between spin and orbital angular momentum has been observed. It is known that electromagnetic waves transport spin angular momentum, which can be transferred to highly birefringent particles [4

] have controlled the orientation and rotation of birefringent particles using a spatial light modulator (SLM) for modifying the polarization and an array of four microlens generating four static optical tweezers. Recently, Preece et al.[8

], much effort has been devoted to extend optical tweezers to the measurement of additional degrees of freedom, such as rotation. Hence, it is of great interest to find methods to control twist and measure torque while simultaneously measuring linear motion and force with the resolution of traditional optical tweezers. The first methods for providing controlled rotation into an optical tweezers were performed by adding a rotating micropipette [10

]. The authors were able to induce twist on a molecule or molecules of DNA while simultaneously measuring force and extension of the molecule.

In this article, we report a new method for generating multiple dynamical optical tweezers, where each one of them is generated with an independent linear polarization state with arbitrary orientation. To codify both the diffractive elements and the polarization states, we make use of the approach described in Ref. [12

], where the phase of the wavefront is modulated in one SLM, and a second one acts as a polarization rotator. However, we have employed a double incidence on a single reflective liquid crystal display (LCD), instead of using two of them. By a slight modification of the experimental setup, circular as well as elliptical polarization states can be generated. We would like to point out that our setup has been designed for a SLM based on nematic liquid crystal cells with parallel alignment (PAL-LCD). In general, PAL-LCD devices differ from TN-LCDs in not having a rotor effect on the incident light when adding a phase. In addition, a technical challenge when designing an experimental setup using PALs devices, compared to TN-LCD ones, is imposed by the constraint of using low incident angles, typically < 6°. Hence, the differences when producing polarization states at individual traps suggest the need to implement different algorithms for generating multiple optical tweezers.

], the laser beam was split into two beams for each half-area of a tilted SLM. The single pass by the phase modulator area adds two holograms with a phase difference S that determines the polarization state. The holograms for multiple traps are calculated by use of algorithm in Ref. [13

], which is based on the extraction of the argument from the complex sum of each hologram of an individual trap. The beams are mixed and the fields with orthogonal directions and local birefringence S are added. The S control permits them the polarization commutation [14

]. In this way, the information of each trap was totally distributed in the whole half-area of the SLM, requiring an exhaustive pixel-to-pixel coincidence between the two modulated beams. On the other hand, the PAL-LCD does not offer the possibility of being tilted. Therefore, we opted for two serial stages: the phase modulation and the polarization rotator. Further, we have a random multiplexing for all holograms of the individual traps [15

]. This strategy allows us to have more tolerance for the coincidence between both the multiplexed hologram and the polarization mask.

Devices based on liquid crystal on silica (LCoS), like we used in the experiment, are very versatile and are easily operated from a computer. However, LCoS modifies the set of aberrations produced by the optical setup, which appear due to the technical limitation when polishing the silica plate to which LC is attached, thus contributing to the primary astigmatism [16

]. Other sources of astigmatism in the experimental setup come from the oblique incidence of the laser beam on the LCD and other optical components. It is known that the aberrations reduce the trap performance metric, diminishing its quality and strength [17

]. Recently, in the search for absolute calibration for optical tweezers including aberrations, it has been reported, in a numerical study, that astigmatism is the primary aberration leading to the strongest reduction of the transverse stiffness [18

], we have employed a simple strategy for estimating and canceling out the astigmatism present in our setup, which is based on the evaluation of symmetry properties of an optical vortex. This method takes advantage of the sensitivity to the phase irregularities of the Laguerre-Gauss beams with low topological charge, recovering the aberration map of the LCD using an iterative method based on the Gerchberg-Saxton algorithm.

This article has been organized as follows: In section 2, we describe the optical setup that allows both generation of multiple traps and control of polarization at individual traps. We also explain the method for correction of primary astigmatism in the whole setup. In section 3, we show some applications of the method for orientation, rotation and displacement of birefringent micro-particles. In section 4 we present our conclusions.

2. Setup description

The experiment for generating holographic optical tweezers (HOT) with independent polarization
control includes several stages, which are shown in Fig. 1. A
collimated and expanded laser beam is used as a source of highly coherent light. In the phase
modulation stage, the wavefront is modified with a combination of diffractive optical elements and
lenses for each trap, by randomly multiplexing the multiple capture points, following the procedure
described in Ref. [15

]. The pixel size of the
random mask contains 10 × 10 pixels of the SLM. Furthermore, a rotation plate, built up
using the same phase modulation device, is used for controlling the polarization at the hologram
areas where the traps are created [12

]. A
telescope couples the laser beam with both phase and polarization information over the microscope
objective with high numerical aperture (NA), thus generating multiple traps at the focal plane of
the microscope objective (or sample plane), where each one of the traps possesses independent
polarization information.

Fig. 1 A schematic diagram of the stages employed for generation of multiple holographic optical traps with independent control of polarization states.

The experimental setup used for the generation of multiple traps with independent control of
polarization is shown, schematically, in Fig. 2. In our case,
the polarization of the laser beam is rotated by means of HWP1 to maximize the transmitted power at
the setup. The beam from a diode laser (Lasiris Coldray laser, 642 nm and 120 mW), with a
transversal Gaussian profile, is expanded up to 5 mm width for an optimal use of SLM in a double
pass of laser beam. The transversal area of a SLM Holoeye LC-R1080, with a PAL-LCD screen and a
pixel pitch of 8.1 μm, was split into sections A and
B with the purpose of carrying out phase modulation and polarization rotation,
respectively. Hence, the P1 polarizer selects the linear polarization of the beam,
which coincides with the director axis of the liquid crystal. The pixel structure of phase mask
addressed in section A is replicated in B with a lens system in a
4f configuration, denoted by lenses L1 and L2.
The P2 polarizer selects the field component along the horizontal direction. The
system for controlling polarization orientation is implemented by a quarter wave plate (QWP1), the
LCD and an additional QWP2. In the next section we will describe this system in detail. The lenses
L3 and L4 form the image of the field in section B of
the SLM in the output pupil of the microscope objective (Nikon 100x, Plan
Apochromatic, NA 1.4). A white light lamp illuminates the sample, a solution with birefringent
micro-particles, and the image is captured by a CMOS camera (Thorlabs DCC1545M). A dichroic mirror
(DM) allows both to maximize the laser power at the traps and to discriminate this light from the
reflected one at the sample holder, which reduces the visibility of the traps. A white light source,
microscope objective, dichroic mirror, sample holder and additional optics are mounted in an
inverted microscope Nikon Eclipse Ti-U.

Fig. 2 Experimental setup with double incidence in a single SLM for both phase (A) and polarization (B) control. An example of masks for the creation of two tweezers with rotated states of polarization allows one to observe random distribution displayed at masks A and B.

Using an interactive program, dynamical manipulation and visualization of multiple traps was performed, including polarization control as well as partial compensation of aberrations. The maximum power measured at the plane of the sample was 18 mW, namely, approximately 15% of the laser power. The minimum power that we tested was 3.1 mW at the plane of the sample and the system still worked. In the following section we explain the procedure employed for the orientation of linear polarization and the strategy used for estimating and compensating the astigmatism in the setup.

]
have shown that a retardation plate can be built up using a LCD-PAL (see the B region
in SLM in Fig. 2) located between two QWPs oriented to a
−π/4 and a π/4 rad in relation to the
crystal director axis. Thus, we can control the phase and polarization of the wavefront using a
double pass throughout a single LCD device. In the Jones formalism, the effective system formed by
this kind of LCD and two QWPs is represented by:

MRot=R(−π/4)(100i)R(π/4)(100exp(iϕ))R(π/4)(100i)R(−π/4),

(1)

where MRot denotes the effective programable
retardation plates depending on the grey level displayed at the SLM.
R(χ) is the rotation matrix and ϕ is
the phase added by the SLM. The Eq. (1) can be
expressed as [12

In the above expression, the direct dependence of the rotation angle of an incident polarization
state on ϕ is clear, allowing an approximated rotation interval of
1.1π rad. Furthermore, an additional phase is added to the rotation, which
must be compensated at A region of the SLM. In the sample plane, the measurement of the Stokes
parameters [20

] was used to estimate the
change of polarization orientation by displaying the gray levels at the LCD. The results are
depicted in Fig. 3. We implemented standard polarimetry for
determining the response of the SLM when used as a programable retardation plate. That is, we add a
QWP and a polarizer at the sample plane, and they were configured for polarization analysis. This
allows us to obtain the Stokes parameters by measuring the power at the sample plane as a function
of the grey level displayed at the SLM.

Fig. 3 Polarization states as a function of gray level displayed at the LCD acting as a rotator. A strong lineal dependence exists between the rotation angle and gray levels. Nonetheless, for intermediate gray levels the polarization ellipses show a slight decrease of their flattening.

Both to verify the degree of coincidence of regions in A and B at the SLM and to show the
rotation of polarization in the illuminated area, a modification of the experimental setup was
implemented (see Fig. 2): A linear polarizer was inserted
between lens L4 and the dichroic mirror, and the images at the SLMs pupil image
(just before the microscope objective) were registered. The expected result is that the limiting
regions of the holograms randomly distributed coincide with the dark zone due to the absorption of
the polarizer where there is no horizontal polarization. In Fig.
4 phase and polarization masks are depicted, where the whole illuminated area of the SLM by
the expanded laser beam can be observed. In this figure, for visualization purposes, we depict whole
masks where each pixel contains 40 × 40 pixels of the SLM. We selected these masks for
visualization purposes only. In the experiments we used mask with pixels of 10 × 10
SML’s pixels and masks with good functioning up to 5 × 5 SLM’s pixels were
tested. However, in our case, to overcame the needs of a high precision alignment we used larger
pixels at the masks. The quality of the masks is enhanced by reducing the size of the pixels,
however this increases the difficulty when aligning masks at regions A and B of the SLM. The
coincidence of the masks ensures that each trap only has the programmed polarization state through
the implementation of the user interface.

Fig. 4 Images of both the phase and the polarization masks at the SLMs pupil, right before the entrance at the microscope objective for generating (a) two and (b) three traps. A linear polarizer was inserted in order to darken zones with rotated polarization. The red bar is the scale length, of 120 pixels of the SLM, approximately 972 μm.

We used the random mask algorithm because it is required that each trap is spatially well defined, where each pixel of the mask has information of a single trap. This allows to make that the image of each pixel of the mask over the corresponding one in the polarization mask, so that polarization of the trap can be controlled by varying the grey level assigned to those pixels. Usually, for holographic optical tweezers the used algorithm is the lens and grating. In this last case the masks are generated as a superposition of the diffractive optical elements from different traps, namely, each pixel has information of all the traps and consequently it can not be assigned a specific grey level for controlling the polarization of a given trap.

] the same conclusion was obtained when the aberrations were compensated in holographic tweezers setup that uses this kind of SLM. Recently, in the search for absolute calibration for optical tweezers including aberrations, it has been reported, in a numerical study, that astigmatism is the primary aberration leading to the strongest reduction of the transverse stiffness [18

]. The reason for that particular aberration states is related to the curvature of silica plate in LCoS devices. Hence, we can focus the SLM auto-compensation in the second order Zernike polynomials. The improvements of aberration state have impact on trap stiffness, although that effect is higher when smaller particles are captured [17

]. This can be done by displaying in the A area of
the SLM several phase maps given by:

φ(r,θ)=mθ+c2−2Z2−2+c2+2Z2+2,

(3)

where (r,θ) are polar
coordinates; m is the vortex topological charge; Z2−2 and Z2+2 are second order Zernike polynomials for primary astigmatism and c2−2yc2+2 are their coefficients, respectively. In addition, in the plane of the
sample (see Fig. 2) a mirror was located for obtaining the
deformed vortex image. The spectral filter of the camera was removed for better acquisition of the
reflection in the mirror. The processing of each image of the LG beam consists in fitting its binary
version to an ellipsoidal geometry, where the eccentricity determine the value of a normalized
quality function. In this way, a zero value of that function means total symmetry of the beam. The
set of measurements allows to generate a quality function surface in terms of c2−2yc2+2 coefficients. From these maps, the parameter combination that minimizes
the quality function value can be found. In addition, a further improvement of the beam quality can
be achieved by exploring the parameter space around the values that minimize the first search.

A test for astigmatism correction was done using the technique above described. Figure 5 (a) shows the quality function surface through an initial set of
values of second order Zernike coefficients. Once the coefficients corresponding to the lowest
quality function value is determined, then a restricted area around these coefficients is inspected
and the new surface is shown in Fig. 5(b). Hence, the minimum
of the second surface gives the coefficients c2−2 (≈ 2.89 rad) and c2+2 (≈ 0.79 rad) that cancel out the astigmatism in the plane of
the sample.

Fig. 5 Surface maps depicting quality function. This function gives a measure of the symmetry properties of the vortex. (b) Zoom of quality function around the region marked with a dashed line in (a).

The beam quality improvement visualized on the LG beams (with m equivalent to 1)
and point spread function (PSF) are shown in Fig. 6. The
presence of astigmatism in the case of non compensated vortex is noticeable, which can be compared
with the results obtained by Kumar Singh et al. in Ref. [22

]. Although the LG beam symmetry was recovered, the compensated PSF
reveals that the higher order aberrations are still slightly disturbing the beam quality, this does
not have a large effect on the generated traps. Also, larger order Zernike polynomials can be
included for further improvement of the beam quality.

Fig. 6 Comparison of the vortex used without (a) and with (b) astigmatism correction. Likewise, aberrated (c) and partially compensated (d) PSFs are shown. The images are the negative version of the acquired ones.

3. Experimental results

In this study, the preparation method, for generating asymmetric birefringent particles, was as follows: We ground an Argopecten purpuratus shell, approximately 20 mg. This grounded powder was resuspended in 1000 μL of Millipore-Q water. We added 100 μL of a solution of 5% Triton X-100 to stabilize the sample. The mixture was strongly agitated using a Pasteur pipet or vortex. To eliminate larger particles, the sample was centrifuged at 100 rpm for 10 minutes. We selected particles of a size of approximately 4 μm, due to the fact that these birefringent particles are non spherical an easy to visualize of both the spatial mobility of the HOT and the torque effects on alignment and gyration when using linear and circular polarization is obtained. Two experiments were performed to verify the functionality of the stages of our proposal. The main reason for using ground powder of the Argopecten purpuratus shell is simply to have non symmetric birefringent particles, which allows us to easily visualize of the control of orientation, as well as the rotations of these particles. For other uses, it is better to employ birefringente vaterite microparticles.

Firstly, we controlled the orientation angle of a single particle using the interactive
application, the results of which are depicted in Fig. 7.
This test is consistent with the characterization of the SLM acting as a programmable retardation
plate, see Fig. 3.

Secondly, we tested the independent control of polarization control. Two particles were trapped
with horizontal and circular polarization. While one particle was rotating, another one was moved
and returned to its initial position. In Fig. 8, the route of
movements is indicated. The rotation of birefringent objects can also be done inserting a QWP after
L4 lens in the experimental setup shown in the Fig.
2. In this case, the trap is programmed with linearly polarized orthogonal states and the QWP
is oriented at π/4 in respect to one of those polarization directions.

Fig. 8 Route of programmed movements for two birefringent particles. (a) The two particles are rotating in opposite directions, while one is kept in a fixed position the other one is displaced (
Media 2). (b) In fixed positions, the same birefringent particles are rotating in opposite senses and later the rotation of the right hand side particle is switched to the same sense of the other. (
Media 3).

4. Conclusions

We have presented a new experimental configuration for the generation of multiple dynamical holographic optical traps, with independent control of linear polarization states at each trap. Furthermore, through a slight modification of the configuration, circularly polarized states can be generated, which are used in rotation of birefringent particles. In this way, we can access two no-simultaneous sets of polarization states: linear with several orientations and another that includes horizontal, vertical, left-handed and right-handed circular states. The experimental setups proposed in Refs. [7

] a complete polarization control for replicate any state (plane, elliptical or circular) has been shown, it requires a third pass by a SLM, which could affect the trap stiffness. Moreover, a simple non-iterative strategy was introduced and applied for primary astigmatism cancellation at the experimental setup. By inserting a quarter-wave plate before the dichroic mirror simultaneously circular or linear (horizontal and vertical) polarization states can be programmed.

Even when the random mask algorithm is less efficient when a large number of traps is generated, the main limitation for the number of traps that can be generated is the laser power. By increasing the power or using an infrared laser instead of the red one, a large increase in the number of traps can be attained. Due to both the large size of the ground powder of the Argopecten purpuratus shell and the wavelength used, there are limitation on the number of trapped particles. However, this number can be easily increased by using vaterite microparticles and an infrared laser. Actually we observed up to six rotating trapped particles, but in such a case it is not possible to observe the orientation features due to the symmetry of the particles.

The ability to create multiple optical traps with independently controlled polarization is of substantial importance for several interesting biomedical applications as well as in microfluidics devices. As demonstrated in Ref. [24

], these multiple traps may be used for a construction of micropumps allowing for simultaneous optical micromanipulation, actuation and sensing. For that purpose, having the traps with independent polarization enables precise measurements of optical torques and forces without having to deal possible interference between the trap beams. When using SLM for the creation of these traps with controlled polarization state, as described in our work, the pump presented in the previously mentioned paper can be vastly expanded and provide guidance on a much larger scale.

Another area of applications where these types of traps will be of importance is in biomedicine. In a recent work by Wu et al.[25

] an optically-based system based on the laser-driven rotation spheres has been developed to control the direction of growth of individual axons. This helps to unravel axonal path-finding which is important in the development of the nervous system, nerve repair and nerve regeneration. By using the rotating particles one can create a localized microfluidic flow generating shear force against the growth cone. This force can be determined quantitatively. Moreover having a HOT system with independent control of polarization could vastly enable a broadened quantitative study of the direction of axon growth.

The development of a holographic optical tweezer setup with independent control of polarization and position is the first step for the evaluation of protein chains. In particular, we are interested in studying the effect of UV radiation on DNA chains. As described in Goodsell in [26

], the UV radiation can produce in the DNA molecules the conversion of Timin-Citozin in Timin dimers. This modification at the molecular structure will affect the mechanical properties, which can be studied using the optical tweezers technique [27

Acknowledgments

This work was partially supported by
CONICYT Grant
PFB0824. A. Arias is grateful to Enlaza-Mundos 2011 mobility program from the Medellin Municipality (Colombia). J.P. Staforelli acknowledges support from FONDECYT 11110145. S. Etcheverry acknowledges partial support from CONICYT scholarships.

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